We study the transmission problem of ballistic electrons in a finite superlattice of rectangular barriers in the presence of a constant electric field. A total transfer matrix is constructed, based on the eigenfunctions of the field-dependent Hamiltonian. An explicit solution is given for the transmission probability scrT for a lattice of \ensuremath{\delta}-function potentials. Numerical results show that, with increasing field strength, transitions occur in scrT at fixed energy, from transmission to gap and back to transmission. In addition to shifts of band edges to lower energies, an almost complete disappearance of lower transmission bands occurs for large fields.